Control Flow Ana 2

7/28/2019 Control Flow Ana 2

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Control Flow Analysis - Part 2

Y.N. Srikant

Department of Computer ScienceIndian Institute of Science

Bangalore 560 012

NPTEL Course on Compiler Design

Y.N. Srikant Control Flow Analysis
http://goforward/http://find/http://goback/


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Outline of the Lecture

Why control flow analysis?

Dominators and natural loops

Intervals and reducibilityT1 T2 transformations and graph reduction

Regions

Topic 1 and part of topic 2 were covered in part 1 of the lecture

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Why Control-Flow Analysis?

Control-flow analysis (CFA) helps us to understand thestructure of control-flow graphs (CFG)

To determine the loop structure of CFGs

Formulation of conditions for code motion use dominator

information, which is obtained by CFAConstruction of the static single assignment form (SSA)

requires dominance frontier information from CFA

It is possible to use interval structure obtained from CFA to

carry out data-flow analysis

Finding Control dependence, which is needed in

parallelization, requires CFA

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Dominators

We say that a node d in a flow graph dominatesnode n,

written d dom n, if every path from the initial node of theflow graph to n goes through d

Initial node is the root, and each node dominates only its

descendents in the tree (including itself)

The node x strictly dominates y, if x dominates y and

x= y

x is the immediate dominatorof y (denoted idom(y)), if xis the closest strict dominator of y

A dominator treeshows all the immediate dominator

relationshipsPrinciple of the dominator algorithm

If p1, p2, ..., pk, are all the predecessors of n, and d= n,then d dom n, iff d dom pi for each i

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Dominator Example

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Dominators and Natural Loops

Edges whose heads dominate their tails are called back

edges(a b : b = head, a= tail)

Given a back edge n d

The natural loopof the edge is d plus the set of nodes thatcan reach n without going through dd is the header of the loop

A single entry point to the loop that dominates all nodes in

the loop

Atleast one path back to the header exists (so that the loop

can be iterated)



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Algorithm for finding the Natural Loop of a Back Edge

/* The back edge under consideration is n d /*{ stack = empty; loop = {d};

/* This ensures that we do not look at predecessors of d */insert(n);

while (stack is not empty) do {

pop(m, stack);

for each predecessor p of m do insert(p);

}

}

procedure insert(m) {

if m / loop then {loop = loop {m};push(m, stack);

}

}


D i B k Ed d N l L
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Dominators, Back Edges, and Natural Loops


D i t B k Ed d N t l L
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Dominators, Back Edges, and Natural Loops


D th Fi t N b i f N d i CFG
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Depth-First Numbering of Nodes in a CFG


D th Fi t N b i E l 1
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Depth-First Numbering Example 1


Depth First Numbering Example 2


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Depth-First Numbering Example 2


Reducibility


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Reducibility

A flow graph G is reducible iffits edges can be partitioned into two disjoint groups,

forwardedges and backedges (back edge: heads

dominate tails)

forward edges form a DAG in which every node can be

reached from the initial node of G

In a reducible flow graph, all retreating edges in a DFS will

be back edges

In an irreducible flow graph, some retreating edges will

NOT be back edges and hence the graph of forwardedges will be cyclic


Reducibility Example 1
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Reducibility - Example 1


Reducibility Example 2


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Reducibility - Example 2


Inner Loops
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Inner Loops

Unless two loops have the same header, they are eitherdisjoint or one is nested within the other

Nesting is checked by testing whether the set of nodes of a

loop A is a subset of the set of nodes of another loop B

Similarly, two loops are disjoint if their sets of nodes aredisjoint

When two loops share a header, neither of these may hold

(see next slide)

In such a case the two loops are combined andtransformed as in the next slide


Inner Loops and Loops with the same header
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Inner Loops and Loops with the same header


Preheader


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Preheader


Depth of a Flow Graph and Convergence of DFA
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Depth of a Flow Graph and Convergence of DFA

Given a depth-first spanning tree of a CFG, the largest

number of retreating edges on any cycle-free path is thedepthof the CFG

The number of passes needed for convergence of the

solution to a forward DFA problem is (1 + depth of CFG)

One more pass is needed to determine no change, andhence the bound is actually (2 + depth of CFG)

This bound can be actually met if we traverse the CFG

using the depth-first numberingof the nodes

For a backward DFA, the same bound holds, but we must

consider the reverse of the depth-first numbering of nodes

Any other order will still produce the correct solution, but

the number of passes may be more


Depth of a CFG - Example 1
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Depth of a CFG Example 1


Depth of a CFG - Example 2
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Depth of a CFG Example 2


Intervals
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Intervals

Intervals have a header node that dominates all nodes in

the intervalGiven a flow graph G with initial node n0, and a node nofG, the interval with header n, denoted I(n) is defined asfollows

1 n is in I(n)

2 If all the predecessors of some node m= n0 are in I(n),then m is in I(n)

3 Nothing else is in I(n)

Constructing I(n)

I(n) := {n};while (there exists a node m= n0, all of whosepredecessors are in I(n)) do I(n) := I(n) {m};


Partitioning a Flow Graph into Disjoint Intervals
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g p j

Mark all nodes as unselected;Construct I(n0); /* n0 is the header of I(n0) */Mark all the nodes in I(n0) as selected;while (there is a node m, not yet marked selected,

but with a selected predecessor) do {

Construct I(m);/* m is the header of I(m) */Mark all nodes in I(m) as selected;

}

Note: The order in which interval headers are picked does not

alter the final partition


Intervals and Reducibility - 1


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y


Intervals and Reducibility - 2
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y


Interval Graphs
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p

Intervals correspond to nodesInterval containing n0 is the initial node of I(G)

If there is an edge from a node in interval I(m) to theheader of the interval I(n), in G, then there is an edge fromI(m) to I(n) in I(G)

We make intervals in interval graphs and reduce them

further

Finally, we reach a limit flow graph, which cannot be

reduced further

A flow graph is reducible iff its limit flow graph is a single

node


Node Splitting
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p g

If we reach a limit flow graph that is other than a single

node, we can proceed further only if we split one or morenodes

If a node has k predecessors, we may replace nby k

nodes, n1, n2, ..., nk

The ith predecessor of nbecomes the predecessor of nionly, while all successors of nbecome successors of the

nis

After splitting, we continue reduction and splitting again (if

necessary), to obtain a single node as the limit flow graph

The node to be split is picked up arbitrarily, say, the nodewith largest number of predecessors

However, success is not guaranteed


Node Splitting Example
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p g p


T1 T2 Transformations and Graph Reduction
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Transformation T1: If n is a node with a loop, i.e., an edgen nexists, then delete that edge

Transformation T2: If there is a node n, not the initial

node, that has a unique predecessor m, then m may

consume nby deleting nand making all successors of n(including n, possibly) be successors of m

By applying the transformations T1 and T2 repeatedly in

any order, we reach the limit flow graph

Node splitting may be necessary as in the case of interval

graph reduction


Example of T1 T2 Reduction
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Regions
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A set of nodes N that includes a header, which dominates

all other nodes in the region

All edges between nodes in N are in the region, except(possibly) for some of those that enter the headerAll intervals are regions but there are regions that are notintervals

A region may omit some nodes that an interval wouldinclude or they may omit some edges back to the headerFor example, I(7) = {7, 8, 9, 10, 11}, but {8, 9, 10} could bea region

A region may have multiple exitsAs we reduce a flow graph Gby T1 and T2 transformations,at all times, the following conditions are true

1 A node represents a region of G2 An edge from a to b in a reduced graph represents a set of

edges3 Each node and edge of G is represented by exactly one

node or edge of the current graph


Region Example
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Control Flow Ana 2